In a large non-reactive pan, heat 1/4 cup of the olive oil over medium heat.Add the onions and sauté, stirring often, until translucent and tender, about 5 minutes.
Add the peppers and celery and sauté, stirring often, until softened, about 10 minutes.Season to taste with salt and pepper.
Add the garlic and cook until fragrant, about 2 minutes.Tie the thyme sprigs and parsley together with kitchen twine.
Add the tomatoes and the herb bundle to the pan and cover with a lid.
Allow the mixture to simmer, stirring from time to time, for about 15 minutes.
In a separate skillet, heat the remaining 1/4 cup olive oil over medium-high heat.
Add the eggplant, and season with salt and pepper.
Cook until lightly brown on all sides, about 8 minutes.
Add the sautéed eggplant to the tomato mixture. Cover and cook 20 minutes.
Meanwhile, in a small bowl, combine the vinegar and sugar, stirring to dissolve the sugar completely.
Remove the vegetables from the heat and discard the herb bundle.
Stir in the vinegar mixture, olives, and capers. Adjust the seasoning.